What's that mysterious curve?

Using the fact that all subtended angles by a chord at points on a circle are equal to a constant angle, and that constant angle is one half the central angle that the chord subtends at the center of the circle, we can construct the centers of two such circles (on both sides of segment $AB$ ), such that the angle that segment $A B$ subtends at these two centers is equal to $(2 \theta)$ , and then we draw the circular arcs that pass through $A$ and $B$ , on both sides of segment $A B$ . Segment $AB$ is not part of the curve. For the location of the circle centers, we note that since $AB$ is a chord in either circle, the centers lie on the perpendicular bisector of segment $AB$ , and are away from the midpoint of $AB$ by a distance of $\frac{1}{2} AB \cot \theta$